Many will have to "shelter in place" , some will have elaborate, remote, well equipped bunkers....most will be somewhere in between. Like a "snow day" it's better to have children occupied in constructive play than running amok!
Those who home school will have most of these suggested supplies on hand, budget conscious can find them at a "$1 Store" or an Education supply store such as "The Learning Castle" or online. I'd suggest as many activities as possible that do not use electricity or batteries, unless you have an electrical source or a charger. (strongly suggested).
- coloring books & crayons
- board games
- blank paper and pencils (pencil sharpener)
- flash cards
- books and story books appropriate for age
- misc. art supplies like colored paper & paste, glue sticks
- pink erasers
- unscented markers
- kid sized furniture or bean bag chairs
- school desk, table or a lap desk
If your shelter has space and you've had the budget to plan ahead...children should have their own space or area with play equipment or structures just for them. The main point of all this is to keep kids from panic...keeping them occupied and having a "in school" experience will help. Depending on the disaster, it may be their future for some time.
Lets talk food. Along with items to keep little minds busy, diet will also be important. Stock kid's favorite foods and snacks...ones that are healthy and require minimal preparation would be best. Power bars, fruit roll-ups, cereals (use powdered milk), trail mix, etc. Long shelf life is important, but as every parent knows, taste is a major issue! Don't forget a good children's vitamin supplement, and any special medications your child may require. If never needed for a disaster, items can be rotated every so often, as required.